Cosmic rest frame, proper distances and comoving distances

I can suggest a couple of FAQ's and papers related to these issue, and give my own take.

My own take first: all notions of distance in general relativity and cosmology are dependent on one's choice of coordinates. The main reason is that to define distance, one must slice space time into space+time.

The standard choice of cosmological coordinates is a cosmological time based on the FRW metric. Physically, one can regard that objects which are moving with the hubble flow have constant cosmological spatial coordinates. Objects "moving with the hubble flow" will see the cosmic microwave background radiation as being the same in all direction, i.e. isotropic. Clocks on these observers (the one's comoving with the hubble flow) measure cosmological time.

Proper distance in cosmology is the distance measured on a hypersurface of constant cosmological time, i.e one can imagine a chain of observers who all measure their distance to the next observer in the chain at some certain instant of cosmological time, and the sum of these distances is the proper distance.. See in particular Ned Wright's cosmology FAQ on this point, and for a few other distance measures that are easier to measure in practice.

Note that this makes the definition of proper distance coordinate dependent, unlike the definition in special relativity. Thus the concept of "proper distance" in special relativity is different from the concept used in cosmology.

Some of the promised papers: Note these links are links to the abstracts, you can view the full text of the paper in several formats (I'd suggest pdf)

http://www.astro.ucla.edu/~wright/cosmo_01.htm Ned Wright's cosmology tutorial
http://arxiv.org/abs/astro-ph/0310808 "Expanding confusion" especially appendix A
http://arxiv.org/abs/0707.0380 "Expanding space, the root of all evil?"